### Abstract

This paper provides a rigorous mathematical study for assessing the dynamics of smoking and its public health impact in a community. A basic mathematical model, which is a slight refinement of the model presented in [F. Brauer, C. Castillo-Chavez. Mathematical Models in Population Biology and Epidemiology. Text in Applied Mathematics. Springer, 2000; G.C. Castillo, S.G. Jordan, A.H. Rodriguez. Mathematical models for the dynamics of tobacco use, recovery and relapse. Technical Report Series, BU-1505-M. Department of Biometrics, Cornell University. 2000], is designed first of all. It is based on subdividing the total population in the community into non-smokers, smokers and those smokers who quit smoking either temporarily or permanently. The theoretical analysis of the basic model reveals that the associated smoking-free equilibrium is globally-asymptotically stable whenever a certain threshold, known as the smokers-generation number, is less than unity, and unstable if this threshold is greater than unity. The public health implication of this result is that the number of smokers in the community will be effectively controlled (or eliminated) at steady-state if the threshold is made to be less than unity. Such a control is not feasible if the threshold exceeds unity (a global stability result for the smoking-present equilibrium is provided for a special case). The basic model is extended to account for variability in smoking frequency, by introducing two classes of mild and chain smokers as well as the development and the public health impact of smoking-related illnesses. The analysis and simulations of the extended model, using an arbitrary but reasonable set of parameter values, reveal that the number of smokers in the community will be significantly reduced (or eliminated) if chain smokers do not remain as chain smokers for longer than 1.5 years before reverting to the mild smoking class, regardless of the time spent by mild smokers in their (mild smoking) class. Similarly, if mild smokers practice their mild smoking habit for less than 1.5 years, the number of smokers in the community will be effectively controlled irrespective of the dynamics in the chain smoking class.

Original language | English (US) |
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Pages (from-to) | 475-499 |

Number of pages | 25 |

Journal | Applied Mathematics and Computation |

Volume | 195 |

Issue number | 2 |

DOIs | |

State | Published - Feb 1 2008 |

Externally published | Yes |

### Keywords

- Equilibrium
- Mild/chain smokers
- Smoking-related illnesses
- Stability
- Tobacco smoking

### ASJC Scopus subject areas

- Computational Mathematics
- Applied Mathematics

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## Cite this

*Applied Mathematics and Computation*,

*195*(2), 475-499. https://doi.org/10.1016/j.amc.2007.05.012